In turbulence modeling, and more particularly in the Large-Eddy Simulation (LES) framework, we are concerned with finding closure models that represent the effect of the unresolved subgrid-scales on the resolved scales. Recent approaches gravitate towards machine learning techniques to construct such models. However, the stability of machine-learned closure models and their abidance by physical structure (e.g. symmetries, conservation laws) are still open problems. To tackle both issues, we take the `discretize first, filter next' approach, in which we apply a spatial averaging filter to existing energy-conserving (fine-grid) discretizations. The main novelty is that we extend the system of equations describing the filtered solution with a set of equations that describe the evolution of (a compressed version of) the energy of the subgrid-scales. Having an estimate of this energy, we can use the concept of energy conservation and derive stability. The compressed variables are determined via a data-driven technique in such a way that the energy of the subgrid-scales is matched. For the extended system, the closure model should be energy-conserving, and a new skew-symmetric convolutional neural network architecture is proposed that has this property. Stability is thus guaranteed, independent of the actual weights and biases of the network. Importantly, our framework allows energy exchange between resolved scales and compressed subgrid scales and thus enables backscatter. To model dissipative systems (e.g. viscous flows), the framework is extended with a diffusive component. The introduced neural network architecture is constructed such that it also satisfies momentum conservation. We apply the new methodology to both the viscous Burgers' equation and the Korteweg-De Vries equation in 1D and show superior stability properties when compared to a vanilla convolutional neural network.

翻译：在湍流建模中，特别是在大涡模拟框架下，我们致力于寻找能够表征未解析亚格子尺度对解析尺度影响的闭合模型。近年来，研究方法倾向于利用机器学习技术来构建此类模型。然而，机器学习构建的闭合模型的稳定性及其对物理结构（如对称性、守恒定律）的遵循程度仍是开放性问题。为同时解决这两个问题，我们采用“先离散后滤波”方法，即对现有能量守恒（细网格）离散格式施加空间平均滤波器。主要创新点在于：我们将描述滤波解的系统方程组，扩展为一组描述亚格子尺度能量（压缩版本）演化的方程。利用该能量的估计值，可通过能量守恒概念推导出稳定性。压缩变量通过数据驱动技术确定，以确保亚格子尺度能量得到匹配。对于扩展系统，闭合模型需满足能量守恒，我们据此提出具有该特性的新斜对称卷积神经网络架构。由此可保证稳定性，且该稳定性与网络的实际权重和偏置无关。更重要的是，该框架允许解析尺度与压缩亚格子尺度之间的能量交换，从而能够实现反向散射。为模拟耗散系统（如粘性流），框架扩展了扩散分量。所提出的神经网络架构还满足动量守恒。我们将新方法应用于一维粘性Burgers方程和Korteweg-De Vries方程，与普通卷积神经网络相比，展现出更优的稳定性特性。